Irreducible representations

Summary information

Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2 , except in the last column, where we consider what happens in characteristic 2.

Trivial representation

The table below describes a one-dimensional (i.e., degree one) representation of the group. The matrices for the representation are uniquely determined, i.e., any equivalent representation must look exactly the same. The characteristic polynomial and minimal polynomial are also uniquely determined, i.e., they must be the same for any equivalent representation.

The trivial or principal representation is a one-dimensional representation sending every element of the group to the identity matrix of order one. This representation makes sense over all fields, and its character is 1 on all elements:

Element

Matrix

Characteristic polynomial

Minimal polynomial

Trace, character value

1

1

1

1

1

1

1

1

Sign representations with kernels as the maximal normal subgroups

The table below describes a one-dimensional (i.e., degree one) representation of the group. The matrices for the representation are uniquely determined, i.e., any equivalent representation must look exactly the same. The characteristic polynomial and minimal polynomial are also uniquely determined, i.e., they must be the same for any equivalent representation.

The dihedral group has three normal subgroups of index two: the subgroup , the subgroup , and the subgroup . For each such subgroup, there is an irreducible one-dimensional representation sending elements in that subgroup to and elements outside that subgroup to .

These representations make sense over all fields, but in characteristic two, they become the same as the trivial representation.

Here is the representation with kernel :

Element

Matrix

Characteristic polynomial

Minimal polynomial

Trace, character value

1

1

1

1

-1

-1

-1

-1

Here is the representation with kernel :

Element

Matrix

Characteristic polynomial

Minimal polynomial

Trace, character value

1

-1

1

-1

1

-1

1

-1

Here is the representation with kernel :

Element

Matrix

Characteristic polynomial

Minimal polynomial

Trace, character value

1

-1

1

-1

-1

1

-1

1

Two-dimensional irreducible representation

The table below describes an irreducible representation of the group of degree more than one. The matrices for the representation are not uniquely determined -- we can choose alternative matrix descriptions by conjugating all matrices by a common matrix. The characteristic polynomial, minimal polynomial, trace (character), determinant, and eigenvalues for the matrices are, however, uniquely determined, since these are invariant under matrix conjugation.

The dihedral group of order eight has a two-dimensional irreducible representation, where the element acts as a rotation (by an angle of ), and the element acts as a reflection about the first axis. The matrices are:

This particular choice of matrices give a representation as orthogonal matrices, and in fact, the representation is as signed permutation matrices (i.e., it takes values in the signed symmetric group of degree two). Thus, it is also a monomial representation.

Below is a description of the matrices based on the above choice as well as another formulation involving complex unitary matrices:

The same character table works over any characteristic not equal to 2 where the elements 1,-1,0,2,-2 are interpreted over the field.

Here is the size-degree-weighted character table, i.e., each cell entry is obtained by multiplying the character value by the size of the conjugacy class and then dividing by the degree of the representation. Note that size-degree-weighted characters are algebraic integers.

Table of matrix entries

This table satisfies the grand orthogonality theorem. Note that unlike the character table, this table is not canonical but rather, for the degree two irreducible representation, depends on the choice of basis.

Representation/element

trivial

1

1

1

1

1

1

1

1

sign with kernel

1

1

1

1

-1

-1

-1

-1

sign with kernel

1

-1

1

-1

1

-1

1

-1

sign with kernel

1

-1

1

-1

-1

1

-1

1

faithful irreducible representation of degree two -- top left entry

1

0

-1

0

1

0

-1

0

faithful irreducible representation of degree two -- top right entry

0

-1

0

1

0

1

0

-1

faithful irreducible representation of degree two -- bottom left entry

0

1

0

-1

0

1

0

-1

faithful irreducible representation of degree two -- bottom right entry

Action of automorphism group

The automorphism group of the dihedral group preserves the trivial representation, the two-dimensional representation, and the sign representation whose kernel is the cyclic group . The two sign representations with kernels and are exchanged by an outer automorphism.

Isoclinism and projective representations

Grouping by restriction to center

Restriction to center as a representation of center of dihedral group:D8 which is isomorphic to cyclic group:Z2 (this determines, essentially, the cohomology class of the projective representation for the inner automorphism group)

List of irreducible projective representations of the inner automorphism group (which is Klein four-group)

Group ring interpretation

Direct sum decomposition

If is any field whose characteristic is not 2, then the group ring splits as a direct sum of two-sided ideals corresponding to the irreducible representations:

More generally, if is any commutative unital ring that is uniquely 2-divisible, then we can write:

Note that the ring of integers does not satisfy the condition for this direct sum decomposition to hold. Instead we need to use the ring (In general, we need to use a ring that is uniquely divisible by all primes dividing the order of the group).

Explicit decomposition and idempotents

We can write:

where are idempotents. These are called primitive central idempotents.

Note that here denotes the identity of the group, and can also be written as since it gives the identity of the group ring.

Relation with representations of subgroups

Induced representations from subgroups

Since the dihedral group is a finite nilpotent group, it is in particular a finite supersolvable group, and hence, it is a monomial-representation group: every irreducible representation can be realized as a monomial representation, i.e., every irreducible representation is induced from a degree one representation of a subgroup. (Point (5) below explains how the two-dimensional irreducible representation is induced).

The trivial representation on the center induces a representation obtained as a sum of the four one-dimensional representations.

The sign representation on the center (which comprises ) induces the double of the two-dimensional irreducible representation of the dihedral group.

The trivial representation on the cyclic subgroup generated by induces a representation on the whole group that is the sum of a trivial representation and the representation with the -kernel.

A representation on that sends to induces a representation of the whole group that is the sum of the sign representations for the other two kernels.

A representation on that sends to (now viewed as a complex number) induces the two-dimensional irreducible representation.

Verification of Artin's induction theorem

Artin's induction theorem states that the characters induced from characters on cyclic subgroups span the space of class functions. Points (2) and (5) cover the case of the two-dimensional irreducible representation. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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